Lenses for 2D planar and curved 3D laser sheets

ABSTRACT

Various lens shapes are able to transform laser beams into laser sheets that can span both two-dimensional (2D) and three-dimensional (3D) space in a controlled manner. The electromagnetic characteristics of the laser beams can be tailored as the laser light enters, traverses, and exits a lens. The projected image can be shaped. The lenses may have sections being multiply or simply connected with or without cavities. The lenses may also have solid or hollow race features, and coatings and/or material layers, which affect the output laser sheet. A fan angle of the produced laser sheet can be up to and include 360 degrees.

CROSS REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of U.S. Provisional Application No.63/006,113, filed Apr. 7, 2020 and entitled “Rings, Discs, Shells, andSolid Volumes As Transformer Lenses; With Associated Devices,” theentire disclosure of which is hereby incorporated herein by reference.

BACKGROUND

The present disclosure relates to lenses, and their shapes, features,and characteristics, that are capable of transforming an incident laserbeam into a laser sheet. Although various laser devices and systems haveutilized hollow and like lenses, these conventional lenses do notproduce controlled laser sheets.

BRIEF SUMMARY

In accordance with one aspect and various embodiments of the presentdisclosure, a device is configured to transform an impinging laser beaminto a laser sheet. The device includes a lens having an exterior walland is configured to transform the laser beam into a 2D laser sheet or a3D laser sheet depending on the alignment of the lens relative to thelaser beam. The lens is configured to transform the impinging laser beaminto the laser sheet having a fan angle up to and including 360 degrees.The lens may also include a race extending therethrough, such as betweenan inlet port at an exterior surface of a solid portion of the lens andan outlet port at another exterior surface of the lens or a terminatingend within the lens. The lens is configured to allow a portion or atotality of the laser beam through the race, from the inlet port to theoutlet port. A lens can include more than one race, a race can behollow, solid, or a combination thereof, and any race may or may nothave an exit port of the lens. Various portions of the lens also may bea dielectric or metallic material, have coatings thereon, and/or includeapertures and/or grids and/or gratings on the exterior surface.

The foregoing and other features of the invention are hereinafterdescribed in greater detail with reference to the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates a first example 2D laser sheet produced by a lens;

FIG. 2 illustrates a second example 2D laser sheet produced by a lens;

FIG. 3 illustrates a third example 2D laser sheet produced by a lens;

FIG. 4 illustrates a fourth example 2D laser sheet produced by a lens;

FIG. 5 illustrates an example 3D laser sheet with curvature produced bya lens;

FIG. 6 illustrates an example simply connected section of a lens;

FIG. 7 illustrates another example simply connected section of a lens;

FIG. 8 illustrates yet another example multiply connected section of alens;

FIG. 9 illustrates an example ring shaped lens;

FIG. 10 illustrates an example simply connected section of a lens withcoating;

FIG. 11 illustrates an example laser sheet produced by a lens sector;

FIG. 12 illustrates a plurality of curve types from which lens sectionsmay be shaped;

FIG. 13 illustrates an example laser sheet output and laser sheetorientation outcome;

FIG. 14 illustrates another example laser sheet output and laser sheetorientation outcome;

FIG. 15 illustrates yet another example laser sheet output and lasersheet orientation outcome;

FIG. 16 illustrates still another example laser sheet output and lasersheet orientation outcome;

FIG. 17 depicts a qualitative overview of the laser sheet output;

FIG. 18 illustrates an example simply connected section of revolutionfor a lens;

FIG. 19 illustrates another example simply connected section ofrevolution for a lens;

FIG. 20 illustrates an example multiply connected section of revolutionfor a lens;

FIG. 21 illustrates another example multiply connected section ofrevolution for a lens;

FIG. 22 illustrates a plurality of example prismatic lens shapes,

FIG. 23 illustrates a plurality of example ring-shaped lenses;

FIG. 24 illustrates a plurality of example shell-shaped lenses;

FIG. 25 illustrates an example multi-shell shaped lens;

FIG. 26 illustrates a plurality of example disc-shaped lenses;

FIG. 27 illustrates a plurality of example plate and solid volumelenses;

FIG. 28A illustrates an example ring-shaped lens having a race formedabout an exterior surface thereof;

FIG. 28B illustrates an portion of the ring-shaped lens of FIG. 28A witha plurality of interior races;

FIG. 28C illustrates another portion of the ring-shaped lens of FIG. 28Ashowing the trace of a plurality of interior races;

FIG. 29A depicts an example toroidal full ring lens with races andports;

FIG. 29B illustrates internal race circuits and ports of the lens ofFIG. 29A;

FIG. 29C illustrates example material layers of a race;

FIG. 30 illustrates an example full ring-shaped lens having bothmetallic and dielectric materials and ports;

FIG. 31 illustrates an example full ring-shaped lens having races andports and metallic internal portions;

FIG. 32 illustrates a generalized example full ring-shaped lens, whichcan be divided into numerous ring sectors;

FIG. 33 depicts a plurality of ring sectors;

FIG. 34A depicts an example ring sector having a plurality of gratingsat an exterior thereof; and

FIG. 34B depicts an example ring sector having a plurality of apertures(or grids) at an exterior thereof.

DETAILED DESCRIPTION OF THE DRAWINGS

Briefly, the following disclosure relates generally to a set of lensshapes, and more particularly to lens shapes to transform laser beamsinto laser sheets. The output laser sheet can span both two-dimensional(2D) and three-dimensional (3D) space, where sheets can have a curvatureand be projected onto a projection surface in a controlled manner. Thiscan be accomplished, for example, because the electromagnetic (EM)characteristics of the laser beams can be altered by the lens. Forinstance, lens shapes can have solid or hollow shaped race featurestraversing the lens exteriorly and/or interiorly that alter the EM waveof the laser beam as it traverses though the lens, and thus affect theoutput laser sheet. Similarly, coatings and material layers, lensshapes, and other design features interior and exterior to the lens, canalso alter the EM wave of the laser beam and affect the output lasersheet.

For example, a fan angle of a laser sheet can range from angles lessthan 90 degrees to an upper angle of 360 degrees. The fan angle isdetermined at least by shapes of lens sections that are simply connectedwithout cavities, simply connected with at least one cavity, and/ormultiply connected. Dielectric and metallic layers can also be stackedwithin the lens. For example, metallic materials can be used for thebase material of the lens with dielectric layers stacked thereon, or themetallic material can be layered within a lens having a dielectricmaterial base. The metallic materials can be imperfect, good, andperfect conductors. Any of the layers can be offset from a base materialof the lens, such as to form a cavity between the base material and alayer over the base material. The layers can also have differentthicknesses from one another. For example, coating designs and layeringdesigns can be uniform or selectively deposited. The coating designs andlayering designs can approach thin films.

In use, lenses described herein that can produce laser sheets have manypossible uses. For example, the laser sheets can be used to determinerange and positioning of objects that have dimensions other than whatconventional radar can detect. In these uses, the lens can be mounted ona spindle similar to conventional radar such that transmitted lightpulses originating from the lens reflect off of an object and returnback towards the source to a receiver/antenna.

In another example use of a laser sheet, a laser sheet transmitter(e.g., a laser and lens that produces a laser sheet) combined with aphoton sensor strip and an electronic signal processing unit can be usedto detect objects piercing through or along the laser sheet. Forinstance, electronic sensing planes can be formed along the boundariesof a playing field. If a sporting ball passes through an EM plane (e.g.,a laser sheet) an interruption in the circuit would indicate that a ballpierced through—or along—the bounding plane. With signal processing, theball location and time of the circuit interruption event can bedetermined. Further, light sensitive nano-fibers or nano-particles canbe manufactured into the surface of the object to reflect the laserlight with unique identifying characteristics. This would establish acertain reflection characteristic not present with possible naturalrandom occurrences. Similarly, a laser sheet can be designed to hoverjust over the playing field to detect where and when a ball interruptsthe curved 3D laser sheet. This technology can be applied to othersporting events and/or other applications where tracking objects withprecision is required.

Similarly, lenses that broadcast laser sheets with 3D curvature can bedesigned to hover on the boundary of surfaces that are curved orcrowned. In these instances again, objects of interest passing throughthe sheet can be detected. This can also be beneficial when building upor removing material with precision. For example, positioning atransmitter and receiver strip at a predetermined height where an upper(or crowned) surface will be allows the surface to be built up (orremoved) until the laser sheet no longer completes a circuit along thereceiver strip. At that point, it can be determined that the surface hasreached the desired height.

Still further, laser sheets can be used to set compressive thermalstresses into large engineered products that have histories of fatiguefailures. Setting compressive surface stresses in regions where fatigueexists requires high powered output to supply the heat. For example, alens made of quartz may have selectively deposited coating patches, withhigh temperature capability, on the exterior surface of the lens. Thissurface feature can promote high power pulsed laser light streams toimpinge on a region of an engineered component that requires thermalstress conditioning. Further, hybrid designs utilizing races within alens can facilitate timed pulse strikes by managing position and wavespeed. Such systems could also include cooling heat transfer components(e.g., fans, heatsinks, and the like) for instances in which the heatcarried by the laser pulse increases the temperature of the lens beyonda predetermined threshold. High power requirements, for deviceapplications like this, can be made from quartz or can require a puremetallic composition.

As noted above, lens shapes can affect the characteristics of a producedlaser sheet. The lenses (and sections of lenses) described herein maytake on any suitable shape, including but not limited to, prismatic,ring, disc, plate, shell, or solid volume shapes, where sections ofthese shapes can be simply connected without cavities, simply connectedwith at least one cavity, or multiply connected. The lens can also be ofone or more materials that permit propagation of EM waves, such asdielectrics, metallics, and composites of both dielectrics andmetallics. The surfaces of the lens can be coated or layered, where thecoating or layering can be uniform or selectively deposited over thesurface area. Apertures, grids, and gratings can be part of these lensdesigns. The surface area can be conditioned by a plurality of etchingtechniques. Closed inner cavities can be partially or fully filled withsubstances such as solids, fluids, gaseous vapors, colloids, or plasmasin order to enhance an electromagnetic wave as it traverses the lens.Race features can be integrated into any lens to control the EMcharacteristics of the laser sheet produced by the lens. The racefeatures can be solid or hollow, where suitable, and can reside on thelens surface and/or on subsurface of the lens. According to theaforementioned features of the lens, the lens can affect EMcharacteristics such as wave speed, group velocity, phase velocity,reflection, transmission, refraction, cutoff frequency, amplitudemodulation, frequency modulation, attenuation, fan angle, wave width,and/or impedance of any laser beam passing therethrough. These featurescan also affect transmittance and reflectance of an incident laser beamas it propagates through the lens. In various embodiments, a lens orlens section can include any suitable combination of the aforementionedfeatures.

Embodiments of lenses according to the present disclosure will now bedescribed with reference to the accompanying drawings. Referring firstto FIGS. 1-4 , these figures depict sketches of a laser source (108,208, 308 408), a laser beam (103, 203, 303, 403), a lens (102, 202, 302,402), a corresponding laser sheet (104, 204, 304, 404), a fan anglerepresented by an arc (100, 200, 300, 400), a projection surface (105,205, 305, 405), a distance d (110, 210, 310, 410) from the lens (102,202, 302, 402) to the projection surface (105, 205, 305, 405), and adistance ƒ (109, 209, 309, 409) from the laser source (108, 208, 308408) to the lens (102, 202, 302, 402).

The output of the laser sheet 104 of FIG. 1 has a fan angle 100 greaterthan 180 degrees, which can reach up to 360 degrees. In order to achievesuch a fan angle, the lens 102 is convex where the laser beam 103 firstcontacts the lens 102, and has a hollow center cavity. In thisconfiguration the distance d 110 is a distance from the lens 102 to theprojection surface 105, and the distance ƒ 109 is a distance from thelaser source 108 to the lens 102. In contrast, the fan angle 200 of FIG.2 extends just over 180 degrees, while the fan angle 300 of FIG. 3extends just under 180 degrees, and the fan angle 400 of FIG. 4 extends90 degrees. The projection of FIG. 2 results from the lens 202 beingprovided at a same distance ƒ 209 as the distance ƒ 109 at FIG. 1 , butbeing generally half of the lens 102 in cross-section. Thus, due to thehollow center cavity, the lens 202 has a convex surface where the laserbeam 203 first contacts the lens 202. The projection of FIG. 3 resultsfrom the lens 302 being provided at a lesser distance ƒ 309 from thelaser source 308 as compared to the distance ƒ 109 of FIG. 1 (andaccordingly a greater distance d 210 to the projection surface 205),while having the same shape as the lens 202. The projection of FIG. 4results from the lens 402 being provided at the same distance ƒ 409 asthe distance ƒ 109 at FIG. 1 , but having only a quarter of the shape ofthe lens 102 in cross-section. The projected image length on projectionsurfaces 105, 205, 305 and 405 can be much greater than the distances dand ƒ shown. Further, increasing the distance d increases the size ofthe projected laser sheet.

The incident angle and strike position at the lens has an influence onhow the reflected rays interact with edges of the lens, and thus affectsthe fan angle. For example, although shown as having a fan angle of 180degrees, the lens 202 may produce laser sheets having fan anglesapproaching 270 degrees depending on the incident angle. Further,coatings applied to the lenses can affect the fan angle. For example, anapplied coating can generate a laser sheet fan angle just below andabove 90 degrees for lens 402, just equal to or above 180 degrees forlens 302, and just above or below 180 degrees for lens 202. Adjustingstrike position and incident angle on lens 202, 302, and 402 can alsogenerate output as shown in FIG. 2 , FIG. 3 , and FIG. 4 . Introducingfacets, or other edge trimming, on edges 222, 333, and 444 can alsocontrol the internal reflections of an incident laser light, and thusthe fan angle of an output laser sheet.

More particularly, an incident angle of 90 degrees (normal) into a raceof the lens can increase the fan angle, where the fan angle furtherdepends on the geometry of the race. Further, incident angles of 90degrees (normal) to the lens surface for simply connected sectionswithout cavities, can produce fan angles less than 90 degrees, while asimilar incident angle for multiply connected lens sections can producefan angles up to at least 180 degrees. Further, in view of Snell's law,there is a critical incident angle at which total internal reflectionoccurs and no leaky refracted light is generated. The effects of strikelocation and incident angle apply to both 2D and curved 3D laser sheets.

As noted in U.S. Pat. No. 10,422,998, issued on Sep. 24, 2019 andentitled “Laser Transformer Lens,” the entirety of which is incorporatedherein by reference, the distances d (110, 210, 310, 410) and ƒ (109,209, 309, 409) are less than the length of the laser output projectionsextending from a center of the projected laser sheet to an edge of theprojected laser sheet. The power level of the laser sources (108, 208,308, 408) should be sufficient enough to excite the laser output shown.The incident angles of the laser beams striking the lenses (102, 202,302, 402) are oblique, and the strike location can influence the outputof the laser sheet, as noted above. That is, the fan angle of a lasersheet can be adjusted less than or greater than 90 degrees, and evengreater than 180 degrees and extending to 360 degrees via the shape ofthe respective lens and the location and/or angle of the respectivestrike location thereof by an incident laser beam.

Different from FIGS. 1-4 , the lens output shown in FIG. 5 has acurvature. FIG. 5 depicts a perspective sketch illustrating a laser beam503, a lens 502, 3D laser sheet 504, projection surface 505, and a 3Dlaser curve 506. The laser source is not shown. As seen in the figure,the beam 503 strikes the lens 502 at a skewed angle. The output lasersheet 504 has a fan angle 500 greater than zero degrees and extendingover 180 degrees up to 360 degrees. Also not shown are the distances dand ƒ, however, similar observations for the planar configurationsstated above hold for the 3D configuration. As can be seen in thefigure, the laser sheet 504 strikes the projection surface 505 with aprojected image 506 that has curvature.

According to Snell's law, as the index of refraction increases for amaterial, the velocity of light passing through the material decreases.Also, as a light ray passes from a low index material to a high indexmaterial, the refracted light ray changes directions tending toward thenormal direction. Conversely, as a light ray passes from a high indexmaterial to a low index material, the ray changes directions tendingaway from the normal direction. In view of this, lens materials havingdifferent indices of refraction can be stacked in a manner to controldirection, wave speed, phase velocity, the amount of reflected andtransmitted light, and/or impedance of an incident light ray, therebyaffecting an output laser sheet.

The light ray mechanics of the present disclosure can be described bythe generalized embodiments of lens sections sketched in FIG. 6 throughFIG. 11 . The ensuing embodiments demonstrate that lens shape andsurface condition can influence the output of lenses. The embodiments inFIGS. 7 to 10 generally correspond to the 2D planar output shown inFIGS. 1 to 4 , whereas FIG. 11 corresponds to the 3D output withcurvature in FIG. 5 . The lens sections sketched in FIG. 6 through FIG.11 are meant to be generalized representations of lens sections that canbe designed into practical lens shapes that are simply connected withoutcavities, simply connected with at least one cavity, and/or multiplyconnected sections with at least one cavity, respectively. In general,the curves defining the section boundaries (inner and exterior) can beone or more of curves, piecemeal combinations of different curves,curves and straight lines, or straight lines, and may form any knownshape (such as circles, triangles, rectangle, and the like) or be anarbitrary shape. Additionally, rings, discs, plates, shells, prisms andother solid volume shapes of lenses and lens sections produce leakylight output—the refracted light output exiting the lens exterior—whichforms the laser sheet. Both 2D laser sheets and 3D laser sheets withcurvature are produced, provided that enough power is available togenerate the leaky light, that the preferred location and incident angleis used to strike the lens surface, and where applicable, coatings areapplied.

The incident light rays (from FIGS. 6, 7, 8, 9, 10, and 11 ) refractwithin the lens interior and the initial refracted ray can reflectbetween the lens boundaries making a reflective wave train. For certainstrike location, incident angle, and laser source power level, thereflected wave train (shown by the double stemmed arrows) can refractleaky laser light outward along the lens exterior. The leaky laser lightcan propagate around an entire lens section—or the leaky laser light canpropagate outward from the entire lens (ring and shellconfigurations)—depending on the orientation of the laser beam relativeto the lens section. For some lenses, both 2D and curved 3D sheets canbe generated depending on incident angle, strike location, and powerlevel. This leaky laser light generates the laser sheet.

For practical shaped lens sections, a characteristic length—orcharacteristic radius (a lens section area divided by the lens sectionperimeter)—had an influence on the lens output for a given laser beamdiameter and power level. As the characteristic radius increasedcompared to a given laser beam diameter, the power level of the lasersource had to increase to achieve the same output as a lens with asmaller characteristic radius. Hence, for a given characteristic radius,the laser power, beam diameter, strike location, and incident angle canbe adjusted to control the leaky light output. Similarly, characteristicradii were found for simply connected sections with one or more cavitiesand for multiply connected sections. From a power perspective, thecharacteristic radii can control laser light source power requirements.

FIG. 6 depicts a simply connected lens section 600 without cavities,bounded by a continuous curve C₁. In the example of FIG. 6 , the lenssection is free from surface conditioning effects and the lens sectionis co-aligned in the same (or nearly the same) plane as the incidentlaser beam. The incident light ray is shown as a solid vector at anincident angle Θ_(i) with respect to the normal direction defined by theunit normal â_(o). A material locating curve, s, is shown that startsfrom the unit normal at the incident location and can stop at anymaterial point along the surface. The refracted rays for the first threerefraction events of the incident light are shown as double stemmedvectors where the 1^(st), 2^(nd) and 3^(rd) generation refracted raysare excited. Normal vectors â₁, â₂, â₃ are shown at the refraction eventlocations. As sketched, the medium propagating the incident ray withinthe section boundary curve C₁ has a lower index of refraction than thelens index, as the refracted wave is oriented towards the respectivenormal directions shown at the incident location. Accordingly, therefracted light passing through the lens exterior—the leaky refractedlight—exits the lens such that the rays are oriented away from therespective normal directions as would be expected for a lens with arefraction index higher than that of the interface material. Also shownin FIG. 6 is an arbitrary point P with a normal direction â_(P) locateda distance s from the incident location. By adjusting at least theincident angle and power level of the laser source, any point P canproduce a leaky refracted light ray for this section.

Similar to FIG. 6 , FIG. 7 illustrates the ray trace for a simplyconnected section 700 with a cavity, the outer lens shape being boundedby curve C₁ and the cavity shape being bounded by curve C₂. Again in theexample of FIG. 7 , the curves C₁ and C₂, are without surface effects(e.g., reflective coatings), similar to the lens of FIG. 6 . Wherepossible, the refraction index of the section is shown higher than themedium which the incident ray propagates through, as is implied by therefraction angles sketched where refraction events occur. Normal vectorsâ₀, â₁, â₂, â₃ are shown at the refraction event locations where theleaky refracted rays exit the lens on boundary C₁. The section isco-aligned with the same (or nearly the same) plane as the incidentlaser beam. Again, by adjusting at least the incident angle and powerlevel of the laser source, any point P along s can produce a leakyrefracted light ray. The light rays contributing to the leaky lightrefraction are excited primarily by the light rays traversing the lensmaterial.

Similar to FIGS. 6 and 7 , FIG. 8 illustrates the ray trace for amultiply connected section 800, having an outer shape bounded by curveC₁ and in inner cavity shape bounded by curve C₂. In this example, curveC₁ is without surface effects while C₂ is completely coated with areflective coating 802. In this configuration, the refraction index ofthe lens material is higher than the medium which the incident raypropagates through, as is implied by the refraction angles sketchedwhere refraction events occur. Normal vectors â₀, â₁, â₂, â₃ are shownat the refraction event locations where the leaky refracted rays exitthe lens on boundary C₁. As seen in FIG. 8 , the reflective coating 802along boundary C₂ prevents leaky refracted rays from exiting into thecavity of the lens section, in contrast to the lens section of FIG. 7 .The section is co-aligned with the same (or nearly the same) plane asthe incident laser beam. Again, by adjusting at least the incident angleand power level of the laser source, any point P along s from theincident location with normal vector â_(P) can produce a leaky refractedlight ray.

FIG. 9 illustrates a generalized ring-shaped lens 900 with an irregulartangential path s and simply connected section 902 without cavities.Both the section and the path which the section is swept about to createthe lens are generalized shapes. The inboard region of the ring 904,defining an orifice through the ring, is coated with a reflectivecoating. The incident ray is shown as a solid vector striking the lensat a location with a unit normal â_(o). The incident angle is such thata planar output is excited in the x-z plane, which is perpendicular tothe plane x-y of the section. A train of reflected light rays, dwellingin the x-z plane, are also sketched. A point P, having distance s fromthe incident location, with normal vector â_(P) can produce a leakyrefracted light ray provided there is sufficient power coming from thelaser source and sufficient incident angle, at least. The battery ofreflected rays are shown along with the leaky refracted rays that formthe laser sheet output. The reflective coating eliminates leaky laserlight from propagating towards the lens center so that maximum leakylight is directed outward from the lens center. Similar to thecharacteristic radii referred to earlier, rings also have acharacteristic length that indicates a power level demand for generatinglaser sheets with various fan angles. This length is found by the ratioof the illuminated area that the light path sweeps out (internal to thelens) by the reflective wave train divided by the perimeter of the areaalong s. Similar length observation can be found for shell shaped lensestoo.

FIG. 10 shows a section 902 of the ring shaped lens sketched from FIG.10 , where the inboard region is covered with the reflective coating1000. The reflective coating 1000 eliminates secondary light effectsreferred to in the discussion of FIG. 7 , where in this instance,secondary lighting effects refer to the lens ring and not the lenssection. The internal light rays are shown sketched in the x-z planerelative to the x-y axes. A leaky refracted light ray is also showncollinear with the x-z plane. The normal vector â_(P) at the arbitrarypoint P along s is included for reference to the inboard region as aportion of the exterior surface which has a radial component of thenormal vector, for a given section, pointing toward the ring center.

FIG. 11 depicts a lens segment 1100 from a generalized ring with atangential path similar to that shown in FIG. 9 . The lens section 1102,of the lens segment 1100, is a simply connected circular shape where theinboard surface region is coated similar to that shown in FIG. 10 . Theincident ray is shown as a solid vector at the incident angle shown, andoriented relative to the lens surface such that the lens emits leakylaser light in the form of a 3D curved sheet 1104. The normal vectorâ_(o), at the incident location is also shown. A battery of reflectedlight rays is illustrated with the leaky light rays forming the curvedlight sheet 1104. In this configuration, the train of reflected raysinternal to the lens is not parallel to the section 1102 and is notperpendicular to the section 1102—the rays travel along a surface paththat is oblique to the section 1102. The curved light sheet 1104 isoutput due to this incident angle and assumed power level and path oftravel within the lens section 1102, and the shapes of the section 1102and segment 1100. This 3D curved output can be generated not only byrings as shown, but also lenses shaped as shells, prismatic shapes andother solid volumes. Impingement angle, characteristic radius, and powerlevel control at least the degree of fan angle output for laser sheetsin general.

FIG. 12 illustrates fifteen (15) different example curves that can beused to form sections of lenses according to the present disclosure.Nearly all of these curves can be represented by infinite power series.These include at least, but are not limited to, the circle, ellipse,hyperbola, and parabola (conic sections), sinh and cosh (hyperbolicfunctions constructed from the exponential function), sine and cosine(harmonic functions), straight line curves oriented along thehorizontal, vertical, and incline, an arbitrary curve, a cycloid curve(e.g., the solution to the Brachistochrone problem), and/or a parametriccycloid (which can be represented by a polynomial). Inverse functionssuch as logarithmic, arcsine, arccosine, arcsinh, and the like, may alsodefine section shapes. Further, any linear algebraic operation can beperformed on these functions to obtain variant forms. These linearoperations can be addition, subtraction, multiplication, and ordivision, and the curves are recognized to satisfy Abelian groupoperations in the commutative sense. These operations can be performedonto one or more functions. For example, for a sine function ƒ1, 1/ƒ1generates the cosecant function. Similarly, if for a cosine function ƒ2,ƒ1/ƒ2 generates the tangent function. And for an exponential functionƒ3, then ƒ1×ƒ3 would be a harmonic function higher order to both ƒ1 andƒ3 individually. Furthermore, the linear horizontal line can beinterpreted as a curve with constant, c, where 1/c or c can be used toshrink or expand each function, respectively. For example, for c equalto ½, multiplying the straight curve into the sum of exp(x)+exp(−x)would return the catenary curve. For c equal to 2, dividing the straightline into the difference of exp(x)−exp(−x) would return the hyperbolicsine curve. Another example of how linear operations can be performedwith these curves is with the sine and cosine curves. If the argument ofboth functions is the same, the sum of the squares of these functionsgenerate the straight horizontal curve. These primitive functions canalso be approximated numerically by the Bernstein polynomials, Basissplines, and non-uniform rational Basis splines (NURBS), which are usedin most contemporary analytical engineering programs.

FIGS. 13 to 16 illustrate a laser sheet output radiating from lenses ofsimilar shapes, where the lenses form shapes between an inner surface S₁and an outer surface S₂. Each surface S₁ is coated with a reflectivecoating. A material position curve s having an arbitrary unit normalvector â_(P) is shown, where â_(P) is located anywhere along curve s.For all cases the power level, incident strike location, and laser beamincident angle is assumed to be such that the output shown can beproduced. Two coordinate systems are provided for reference in thefigures: a first in the ζ, η, λ space, and a second in the x, y, zspace. The ζ, η, λ system tracks the lens section orientation while thex, y, z space tracks the laser beam orientation. These systems emphasizethat each figure has at least two possible configurations capable ofgenerating 2D laser sheets. The first case corresponds to lensorientations where the lens section shares the same plane as the laserbeam such that ζ=x and η=y. The second case corresponds to lensorientations where the lens section is in the ζ=z and η=yplane—perpendicular to the x-y plane of the laser beam. In other words,for the first case, the laser beam and the lens section share the sameplane of the laser sheet output, whereas for the second case, the planeof the lens section is perpendicular to the plane spanned by the lasersheet. The lens shapes used are uniform, with material removedsequentially to capture the different lens output as the material isremoved. Other shapes could also be used to demonstrate the effect,however, using these shapes showed how derivatives of a single shapeproduces different controlled outputs.

For the first case, the lens section resides in the ζ-η plane andradiates a 2D laser sheet in the ζ-η plane while the laser beam is inthe x-y plane. As shown in FIGS. 13 to 16 , the fan angle output is 360,270, 180, and 90 degrees, respectively, provided that the laser beam hassufficient power level, incident angle, and strike location. The ζ-ηplane shares the lens section and the x-y plane of the laser beam. Thenormal vector â_(P) could slide to any point P along the exterior of thelens. For FIGS. 13 to 16 , â_(P) could slide tangentially over 360, 270,180, and 90 degree tangential sweeps, respectively, thereby allowing theleaky laser light to form a laser sheet over the tangential rangesabove. Prismatic lens configurations are an example that has thissection orientation.

The second case applied to rings and shells where a ζ-η plane passesthrough a lens formed into a full ring or full shell (FIG. 13 ), andsectors thereof (FIGS. 14 to 16 ), while the laser beam is in the planedefined by the x-y coordinates. The lens section is in the ζ-λ plane,where the section is swept about a revolution axis in a direction alongη (the axis of revolution being parallel to λ, and η being the directionthat the section is swept about) with a constant radius. For sufficientpower level, incident angle, and strike location, this configurationradiates a 2D laser sheet with a fan angle of 360, 270, 180, and 90degree outputs, corresponding to the lenses shown in FIGS. 13 to 16 ,respectively.

For both the first and second cases, the common elements of theseconfigurations are an inner and outer wall for the reflective wave trainto interact with, a reflective coating on the inboard region, and amedium for the wave to traverse through. The normal vector â_(P) can bepositioned at any lighted point along the exterior over a 360, 270, 180,and 90 degree tangential sweep for FIG. 13 through FIG. 16 ,respectively. Provided leaky light rays pass through all of the pointsat any â_(P), a laser sheet would be launched as shown. The initial andterminating points—and all points in between—that â_(P) can be locatedat, govern the fan angle of the light output. For example, if the lenssurface S₂ in FIG. 13 is coated with a reflective coating (r=1) over alit portion of S₂ then the leaky light rays cannot pass thru the coatedregion hence reducing the full at least 360 degree fan angle to that ofa lesser fan angle.

By orienting the laser beam in a plane that is oblique to the x-y, x-z,and y-z planes, 2D laser sheets and 3D laser sheets can be excited. Theshape of the bounding surfaces S_(n), characteristic radius, powerlevel, incident angle, and the strike position of practical lensconfigurations determine the shape of the leaky light output. For alllens configurations, a limiting value of the incident angle would bethat angle which promotes total internal reflection, where no refractedlight leaks out from the lens surface.

For both the first and second cases, the lenses are closed volumesbounded by surfaces S₁ and S₂, and it is the bounding surfaces thatpropagate the reflected light rays. In the limit, it is possible toconstruct lenses with hyper-thin walls (e.g., such as the thickness offoil, about 50 mils or less) that approximate surfaces where acollection of such surfaces S₁, S₂, . . . S_(n), which form a closed oropen chute, could be aligned so that the reflective wave train producesthe desired leaky refracted light ray output. This configuration wouldapproach that of hyper-thin shells and dielectric coatings similar tothe Dupin shells of FIG. 25 (discussed below). A single hyper-thin shellcan have two surface sides to generate a reflective wave train thatproduces the leaky refracted light. Hyper-thin shells can also beenveloped to generate a reflective wave train that produces the leakyrefracted light. In some embodiments, wall thickness of a lens may benon-uniform. For example, as shown at FIG. 33 , the wall thickness oflens 3310 is tapered to form a slot (or chute) through the lens.

For FIG. 13 through FIG. 16 , as the normal vector â_(P) is moved alongan uncoated/unconditioned exterior surface that increases in length, s,the fan angle increased provided that the preferred inputs generatedleaky refracted light. As the unit normal â_(P) is moved until reachingits starting point, without doubling back on curve s (as depicted inFIG. 13 ), the fan angle of light is a 360 degree output. Similarly,where â_(P) begins and terminates for FIGS. 14 to 16 , the fan anglematches the angle traversed by path s—generating 270, 180 and 90 degreefan angle outputs, respectively.

It is lenses with topologies having these characteristics made frommaterials capable of transmitting and controlling EM waves which are ofinterest herein. The materials can be dielectric, or a composite of bothdielectric and metallic. For example, it is conceivable for a prismaticrod, with hemispherical ends, to generate a 360 degree fan angle outputbecause the unit vector â_(P) can travel around the exterior surface, inplan form, and end up at the material curve origin while the reflectedlaser wave train could traverse the lens, in plan form, generating leakyrefracted laser light output. The laser source power level, thecharacteristic radius, strike location, and incident angle influence theoutcome of the leaky laser light output. Again, the output from theoblique orientations can be 2D planar or 3D with curvature. Coatings andother surface effects can be implemented to enhance and tailor theoutput. For example, masking half of the exterior of the lens from FIG.13 with 100% reflective coating would limit the output to that generatedfrom the lens shown in FIG. 15 . It is recognized here that the laseroutput fan angle can be controlled to approach zero degrees, where shapeand/or coating configurations can lead to this limiting output inaddition to power, strike, shape, and/or incident settings.

FIG. 17 provides a qualitative overview of a laser sheet output from anexterior lens boundary, in terms of the underlying mathematical physics.For a preferred power level, strike location, and incident angle, thegeometry of the light output striking a projection surface is shownempirically to be influenced by several parameters including at leastlaser beam diameter, relative distance between the lens surface and theprojection surface, lens shape, lens material, the condition andconditioning of the lens surface area, and the threshold light justpassing across a lens surface, where the laser source power level, laserbeam incident angle, and strike location influenced the latterparameter. An arbitrary 3D surface area patch of a laser sheet 1700 canbe denoted as Ω_(P) bound by a closed curve ∂Ω_(P), and an arbitrary 3Dsurface path of a lens 1702 can be denoted as Ω_(L) bound by a closedcurve ∂Ω_(L). Both light patches Ω_(L) and Ω_(P) are subsets of the litsurfaces on both the lens and the projection surface, and represent asubset of total leaky light output from the lens. For each lens, andcorresponding output configuration, a unique projection is emitted ontoa projection surface. Accordingly, a mapping functions M exists whichcan be used to predict the unique projections, where M is generallythree-dimensional.

For a generalized cylindrical lens shape, the function can be expressedas M=M(m₁(r, θ, z), m₂(r, θ, z), m₃(r, θ, z)). With two mathematicaloperations involving M, the patch size of projected light area Ω_(P) canbe predicted. Consider FIG. 17 where a generalized lens shape is shownalong with the cylindrical <r, θ, z> coordinate system. As noted above,a bounding curve ∂Ω_(L) encloses an areal patch Ω_(L) of emitted lightover a portion of the total lens surface area enclosing lens 1702. Everypoint on Ω_(L) is assumed to emit leaky laser light, where Ω_(L) is alit 3D surface area patch out of the total lens surface area. An arealpatch of projected light Ω_(P) 1700 of a portion of a laser sheet isbound by curve ∂Ω_(P) in a rectangular coordinate system <t, u, v>relative to a projection surface. The mapping function M is representedas light rays with origins located at the surface points residing onareal patch 1700, which are terminated on the laser sheet patch ofpoints bounded by curve ∂Ω_(P) on a projection surface. In other words,for every patch of light on Ω_(L) a corresponding patch of light onΩ_(P) is assigned, based on M. The rays representing M are shownparallel to one another but it is understood they can be randomlyoriented as well.

Operating the Jacobian transform on M provides the terms for theintegrand of the area integral found in Stokes' Theorem. For a fixed setof parameters, these mapping functions can be utilized to predict thesize of the projected image Ω_(P) on the projection surface for anyΩ_(L). In theory, closed form prescriptions of M can be written if it isknown how ray tracing and the identified parameters above relate to eachother functionally. In this configuration one would operate thecylindrical Jacobian transform on M to get an ordered set of functions.These ordered functions are permutations of the products of slopes, anddifferences thereof, which carry physical meaning. These slope productsand differences thereof can be grouped, while preserving the physicalsignificance, into the corresponding expressions of the area integralfound in Stokes' Theorem, hence allowing the prediction of the size ofthe areal patch Ω_(P).

In practice, explicit closed forms of M are scarce, however, M can beconstructed automatically through the use of engineering softwareassuming that the functional relationships discussed earlier were knownand programmed into modules. Numerical fusion of these mathematicalrelationships make the construction of M tractable such that the sizeand shape of Ω_(P) can be designed appropriately. Surface designs onboth Ω_(L) and Ω_(P) can be made experimentally or with engineeringtools to meet service and performance requirements. These toolstypically leverage Bernstein polynomials, and polynomial curvesrepresented by Basis splines, and non-uniform rational Basis splines(NURBS), which can approximate the shapes of the surface bounding thelens sections constructed from the curve set shown in FIG. 12 andnumerous others as well, not shown in FIG. 12 . Euler, MacLaurin, andothers have shown that most of the curves represented by the functionsshown in FIG. 12 , and other functions, can be expressed in terms ofpower series expansions.

By expanding the curves like those from FIG. 12 and others into powerseries format they are classified as polynomials. As the number of termsin the power series grows, for a given curve, the summation of theseries expansion approaches the closed form function of the curve. Itcan be said that the curves from FIG. 12 are a subset of the collectionof Bernstein polynomials and that the Bernstein polynomial set, as wellas polynomial curves generated by Basis splines, NURBS, I-splines,M-splines, and T-splines can define lens sections and surface shapes.

These mathematical formulations assume perfect curves, areas, andvolumes; however, manufactured lenses approach perfect geometries nearlywhile the materials have imperfections through the volume and onbounding surfaces. These imperfections influence the output and can beminimized with manufacturing control processes. Surface conditioning canbe applied to tailor the lens output depending on the servicerequirement. Other additions to the curve descriptions found on FIG. 12that can appear on manufactured lenses include but are not limited tosimple and compound fillets which would smooth the idealized sharp edgesand pointed vertices at cusps presented herein.

As suggested above, lenses can be in the shape of prisms, ring sectors,full rings, discs, plates, shell sectors, shells, and solid volumes.Moreover, all lenses can have race features that are solid or hollow.These races can have shaped entrance and exit ports and other featuresthat enhance the laser sheet output. Samples of these lensconfigurations are described below.

Generalized sections, which can be used to form prisms, rings, discs,plates, shells, and solid volumes, and where applicable sectors/sectionsthereof, are presented next. These shapes have the geometric constructsthat produce 2D planar and 3D curved laser sheet output similar to thatdetailed in FIGS. 11 and 13 to 16 . Sections can be formed by chainingseveral curves, and segments thereof, of the same type, or they can beformed by chaining one or more different curve types together. It isnoted again that all of these curves and curve segments can beapproximated as closely as desired by Bernstein polynomials. Splines canalso be used to construct the curves graphically. The materials used forthe base material of these lenses can be dielectric or a composite ofboth dielectric and metallic.

FIG. 18 illustrates a simply connected section 1800 of an arbitrary discbounded by four arbitrary curves with beginning and ending points 1, 2,3, and 4. A vertical axis defined by points 5 and 6 is shown along withan angle ω. The angle ω indicates that an arbitrary section could bespun about the vertical axis to generate a disc sector or a full disc.Discs can be identified as lenses with diametric length dimensions thatare greater than the thickness dimensions. For thin discs, one couldimpose a limit to the ratio of these dimensional components as 1/10, orlarger ratios for thicker discs. Lenses as plate shapes, on the otherhand, can have the same aspect ratio requirement for a section, however,the diametric component would be replaced with geometries that were notdiametrical—for example, polygons or kidney shapes with thickness.

Similar to FIG. 18 , FIG. 19 also illustrates a simply connectedarbitrary lens section 1900 defined by four curves C₁, C₂, C₃, and C₄with beginning and ending points 1, 2, 3 and 4. Four axes are shown withfour angles ω₁, ω₂, ω₃, and ω₄. Spinning the section about the axesdefined from points 5-6 or 7-8 over angles ω₁, and ω₄ will generateshells which are open on the ends of the section. Depending on the angleused to spin the sections, the lenses can be shell segments or fullshells. Spinning the section about axis 4-5 generates a full shell witha closed end if ω₂ is 2π, and a segmented shell closed on one end forω₂<2π. Spinning the section about axis 1-4 generates a full shell withclosed ends if ω₃ is 2π or a segmented shell closed on both ends forω₃<2π. If aspect ratios of the arbitrary section shown approach one, theshell turns to a ring when the section is spun about axis 5-6. For ω₁<2πring sectors are formed, for ω₁=2π full rings are formed. From atopological vantage point, lenses formed into open full shells and fullrings could be construed to be multiply connected. In a strict sense,for these shapes a theoretical loop cannot be pulled closed withoutleaving the surface. Accordingly, these lenses are not simply connectedalthough the sections shown here are simply connected.

The content of FIG. 20 and FIG. 21 is similar to that of FIG. 18 andFIG. 19 , respectively, except that the sections are multiply connected.That is, FIG. 20 illustrates lens section 2000 having the same shape assection 1800 of FIG. 18 but with cavities C₅, C₆, and C₇, and FIG. 21illustrates lens section 2100 having the same shape as section 1900 ofFIG. 19 but with cavities C₈, C₉, and C₁₀. The cavities in thesesections (C₅, C₆, C₇, C₈, C₉, and C₁₀) can represent local features,periodic features, or a-periodic features that vary with angle ω_(j)(j=1, 2, 3, 4)—or they can represent sections of continuous tubes(races) for laser light to traverse through, where the tubes can splitat one location and rejoin at another location. The inner curve sets C₅,C₆, C₇, C₈, C₉, and C₁₀ contained in these sections can be smooth curvesor straight curves with discontinuities or a combination of both. Thesmooth curves can have cusps, however, it can be argued thatmanufactured products do not have infinitely sharp edges. One or morecavities can be included, and these cavities can be filled (partially ortotally filled) with other materials. Here, it is understood that thecurve label Cn can refer to a group of curves chained together bycoincident end points. Also, the number of cavities in FIG. 20 and FIG.21 is depicted as three (3), however, there can be 1, 2, 3 or more ofthese cavities in a multiply connected section where suitable.

Next, various lens shapes capable of providing a curved 3D laser sheetare described.

Examples of lens shapes with simply connected sections having definitebounding curves are provided next at FIG. 22 . The sketches demonstratea few of the many possible configurations that can produce a laser sheetoutput in 2D planar and 3D with curvature forms. These shapes are merelyexamples of how the above-discussed primitive curves can be used todesign lens sections. These examples are not exhaustive; however, theyprovide samples of the essential shapes which are capable oftransforming a laser beam into a sheet—where the fan angle can spanangles from zero to 90 to 180 to 360 degrees. Sections can be formed bychaining several curves of the same type, and segments thereof together,or they can be formed by chaining one or more different types together.For example, prism 2210 includes multiple sections providing a largefillet feature i, a small fillet feature ii, an ellipse feature iii, anda circle feature iv. In other examples, prism 2206 includes a sinhfeature v; and prism 2214 includes a cycloid feature vi.

That is, the prismatic lenses illustrated include examples swept fromsections that are simply connected without a cavity, simply connectedwith a cavity, or multiply connected sections, and in many examples,where the prism length is much greater than the longest section length.Prismatic lenses 2200, 2202 and 2204 have simply connected sections thatare circular, elliptical, and an 8-sided polygon, respectively.Prismatic lens 2206 is a simply connected section with a cavity that hasfive straight sides on the exterior, four straight sides on the inboardregion, and a curve segment defined by sinh, where the inboard andoutboard regions are joined by two straight curves at two locations. Thelast four prisms are formed from sections that are multiply connected.Prismatic lens 2208 has a section with three straight curves on theexterior and a circular curve on the inboard region. Prismatic lens 2210has a section defined by five straight curves each joined by fillets onthe exterior, and the inboard region is bounded by a circular curvesegment and an elliptical curve segment. Prismatic lens 2212 has acircular exterior with a seven sided polygon for the inboard region.Prismatic lens 2214 has a section with an elliptical inboard region anda ten sided polygon on the exterior that is truncated by a 3D cycloidsurface, where multiple curves are chained together bounding thetruncated surface on the exterior.

In general, the aforementioned prismatic shapes show examples of howcurves and curve segments can be used to construct lenses or lenssections where the curves are of one type, and with more than one curvetype changed together. They also demonstrate that simply connectedsections with and without cavities have exterior surfaces that aredefined with straight curves and curve sections illustrated in FIG. 12 .Also demonstrated are multiply connected sections that have both inboardand outboard regions defined by both straight curves and other curvesfrom FIG. 12 . The fillets, chamfers, bevels or otherwise rounded edgesor corners introduced are expected to be on all lens shapes produced, tosome degree, due to manufacturing processes.

The curves and curve segments discussed above with respect to FIG. canbe approximated by Bernstein polynomials, with uses and restrictionsimposed on them by convention (the resulting curves due to the imposedrestrictions on Bernstein polynomials are known as Bezier curves). Thisrecognition is founded primarily on Weierstrass' Approximation theorem.A 2D generalization can be found in the Stone-Weierstrass theorem wherethe compact Hausdorff space is considered. Furthermore, Runge's theorem,related to the issue, uses compact sets composed of complex numbers toapproximate curves bounding multiply connected regions. Moreover, theliterature regards Mergelyan's theorem as the ultimate generalization ofWeierstrass' approximation theorem and Runge's theorem, and showed thecomplete solution of the classical problem of approximating curves withpolynomials using complex analysis. The set of splines are used incommercial engineering analysis software which employs polynomials toconstruct curves and curve surfaces. These splines are referred to asBasis splines, B-splines, and Non-uniform Rational B Splines (NURBS),where NURBS are an extension of the B-spline. These spline functions arepiecewise polynomials that can be continuous and also have continuousderivatives. These bounding curves can be designed to generate boundingsurfaces to manufacture lenses. Other splines schemes include I-splines,M-splines, and T-splines and other spline schemes not listed here.Lenses with definite shapes bounded by curves forming simply connectedsections without cavities, simply connected sections with cavities,and/or multiply connected sections can have base materials that aredielectric or a composite of both dielectric and metallic materials.

Various full or partial ring-shaped lenses are next shown in FIG. 23 .From a topological perspective, full rings can be thought of as multiplyconnected solids with sections that may or may not be simply connected.The first full ring lens 2300 has a section defined by an ellipse. Thesecond full ring lens 2302 has a section defined by hyperbolic curves onthe surfaces with the inner most radius and outer most radius, wherethese two curves are connected by a parabolic segment on top and astraight curve on the bottom, and where fillets close the section inbetween the curve set. The third full ring lens 2304 has a sectiondefined by a hyperbolic curve on the inner radius and an ellipse on theouter radius, where these curves are joined by two straight horizontalcurves. A Dupin cyclide is presented as a solid ring lens 2306. ThisDupin cyclide is a canal surface having two directices—one hyperbolicand the other an elliptic. This cyclide has circular sections with radiithat vary along the tangent direction. The directices are shown atpartial ring lens 2308. The Dupin cyclide demonstrates how the sectioncan vary with the material curve s.

Ring lens 2310 is an open ring segment having a trapezoid section thatis swept about the arc of a cycloid. Cycloid curves come from the curveset known as Roulette curves, where a Roulette curve is generated by thetrace of a point on a circle as it rolls over another stationary curvewithout slipping. For the case of a cycloid, the other stationary curveis a straight line. Other Roulette curves, in addition to that from FIG.12 , within the scope of this disclosure are epicycloids, hypocycloids,and involutes. Cycloid paths, and other curves, and the curves from FIG.12 , can be used to sweep sections about which define rings, and shells.The cycloid curves can also be used to define lens sections and racesections.

Moreover, it is possible to chain prismatic segments together so thatthey too wrap around a common axis producing closed ring shapes that areknown as toroidal polyhedrons. One such example is shown by ring lens2312 of FIG. 23 , where the truncated prismatic ends that coincide fitflat at the interfaces.

Further, the curves of FIG. 12 , and others, can be used to boundsections of shell lens, for example, as illustrated in FIG. 24 . Theseshell lenses can be centrally open, fully closed, and/or have an openend, where curves, such as a hyperbolic or parabolic curve can defineinner or outer surfaces. In some embodiments, upper and/or lowersurfaces can be truncated to provide flat surfaces. In some embodiments,inner or outer surfaces can be at least partially defined by anexponential function, by a straight inclined curve, and/or by curvesmultiplied together. For example, shell lens 2400 has a parabolicexterior and hyperbolic interior. Lens 2402 has one open end with aninterior surface defined by inclined straights, and an outer surfacedefined by an exponential curve, and a flat on the upper most surface.Lens 2404 has one open end defined by a fillet, and a lower portiondefined by two curves multiplied together exp(x)×sin(x), one offset fromthe other. Lens 2406 has hyperbolic sides with closed ends. Shell lens2408 has an inner surface defined by the cosine function, and an outersurface defined by a sine function. Lens 2410 is defined by a flat top,elliptical inner surface, and parabolic middle exterior portion withflat edges above and below. Finally, lens 2412 has a straight flat topedge with inner and outer surfaces defined by cycloids one offset fromthe other.

FIG. 25 depicts a particular embodiment of a multi-shell lens 2500having a shape of a Dupin cyclide, where the shell thicknesses arehyper-thin. The three spokes, extending from a respective inner shell toa respective outer shell, positioning the shells relative to one anotherare not shown. A portion of two of the shells is removed to show theinternal shapes enveloped within. The inner most shell 2502 is a full360 degree closed shell, whereas the outer shells 2504 and 2506 areexamples of how the Dupin cyclide can be either a full shell or a shellsector. Thin shell thicknesses can assume values up to 1/10 the radii ofa given section. Shell thicknesses larger than that are known as thickshells. The figure demonstrates how the circular sections change alongthe tangential direction and how the Dupin cyclide can be parameterizedto assume many similar shapes. The elliptical directrix and the pointswhere the hyperbolic directrices pass thru the plane of the origin alsoare shown. This sample shape demonstrates how hyper-thin walls can beused to construct lenses having limiting bounding surfaces. In thelimit, hyper-thin walls can approach thicknesses of coatings or foils;however these limits are extreme; and practical hyper-thin wallthicknesses are assumed to be less than 50 mils.

In addition to shell-shaped lenses, disc-shaped lenses can be formed. Insome embodiments, a lens thickness can be smaller than the minimumin-plane dimension. Some example disc-shaped lenses having simplyconnected sections without cavities are illustrated in FIG. 26 . Forexample, disc-shaped lens 2600 has a section defined by chaining acycloid and parabola with two fillets between the two curves. Lens 2602is defined by a section with an outer most radius curve forming acatenary along with two straight curves that formed flat upper and lowersurfaces on the lens. Disc lens 2604 has upper and lower surfacesdefined with sine and cosine curves that form a cusp at the disc centeron the upper surface. Lens 2608 has upper and lower cusps, the sectionformed by intersecting cosh and sinh curves with the Laguerre function0.5(x²−2x+2) and sweeping the section about an axis passing through thecusps. Again, if the angle of revolution ω is <2π the lens would be adisc segment, and if the angle of revolution equals 2π the lens would bea full disc.

Lenses shaped as plates, similar to discs, have thicknesses that aremuch smaller than the smallest plan-form dimension shown here where thesection is not revolved about a common axis. Plates with triangular andpolygonal plan-forms, such as discs 2700, 2702, respectively, as seen inFIG. 27 can be formed. In some embodiments, lenses according to thepresent invention can have solid shapes as also seen in FIG. 27 . Theselenses may have shapes corresponding to spheres 2704 (having a metallicinner core and outer dielectric layer), truncated cone 2706, shape 2708with a section defined by a chain of a straight horizontal, a fillet,and an ellipse divided by a parabolic curve, and hyperboloid 2710, or asolid 2712 with a section composed of a cycloid curve and a horizontalstraight curve. In some embodiments, a lens, such as a solid shape lens,can be composite, such as having a metallic interior enclosed by adielectric external layer.

FIGS. 28A-C illustrate race features integrated into a circular ringwith a generalized section. The solid race also has a generalizedsection. These features can also be incorporated on prisms, plates,discs, shells, and solid volumes, among others. Race features can besolid or hollow. Similar to lens sections, race sections can be derivedfrom the curves shown in FIG. 12 among other curves that can bedescribed by Bernstein polynomials, Bezier curves, and a plurality ofsplines, and the like.

Turning first to FIG. 28A, a race 2800 is a solid with a generalizedsection swept along a plane that has a normal direction oblique to thez-axis, and is embedded into a toroid surface 2802 of the lens 2804. Thematerial of the race 2800 can be a dielectric, different than the basematerial of the lens 2804. The base material of the lens 2804 can bemetallic or dielectric. FIG. 28B shows a general section of lens 2804,with races configured from polygon shapes. Races 2806 and 2808 arehollow internal to the lens boundary while the third race 2800 is solidand integrated to the surface of the lens as described above. FIG. 28Cillustrates the lens material being transparent to enable a view of howthe races 2800, 2806 and 2808 are swept about planes that are obliqueand parallel to the respective lens 2804. Regarding race ports, orinlets and outlets from the main lens body, such ports can be shapedwith uniform cross sections or variable cross sections. It will beappreciated that such solid and hollow race features can be integratedinto any suitable shape and can traverse lens sections of differentshapes. The races may also be of any dielectric material and/or containcoatings and/or layerings already discussed. The ring shape was chosento present race and associated features due to geometric simplicity.Further, races may split into a plurality of paths and reform into asingle path, and separate races may merge, at junctions. In someembodiments, races may be isolated from each other.

In some lens embodiments, the race may be of one or more materiallayers, such as one or more dielectric layers, with refraction indicesdifferent than the base material, one or more metallic layers, or one ormore coating layers, or some combination thereof. The material layerscan have different thicknesses, and can be of different or the samematerials. The layers may be immediately adjacent to each other (e.g.,stacked) or separated by any distance according to a direction ofpropagation of light through the race. A material layer can be offsetfrom the terminating end so as to form a cavity between the offset layerand the terminating end. This offset layer can represent one or morematerials in a layered stack with variable thicknesses. Forconfigurations where the material layers are conductive, the layers ofthe stack on the termination end can be of the same material.

FIG. 29C shows an example of this arrangement. Therein, a hollowone-quarter-turn race 2914 having a rectangular section includes alayered material stacking sequence 2916 at a terminating end 2918, and aconductive material layer 2920 offset from the terminal end 2918. Theangle shown is parametric and can assume numerous values to orient theterminal end relative to the incident wave. The distance d, locating theoffset layer is a quarter of the wavelength for managing wavereflection. Anechoic coating schemes can achieve the same EM waveoutcome while avoiding manufacturing issues associated with both theoffset distance tolerance and angling for metallic layering.

When the material stack at the terminating end is composed of severaldielectrics the layer offset from the terminating end can be a two-plydielectric with different thicknesses. Such layering schemes andorientations can manage EM wave impedance and phase velocity. Layeringschemes can also manage the outcome of the incident wave just as itstrikes a layer surface.

For such ring lenses with one or more races, a race can excite lasersheets along the length of the race. Ignoring any end effects, for thoseraces shown in FIG. 29B the maximum fan angles are 270 degrees for therace with the terminal end and 180 degrees for the race with both inletand exit ports. Depending on coating deposits and the settings of theparameters discussed above the fan angles can vary. Races can enable aportion of or the total laser beam input to enter a race inlet, totraverse an internal geometry of a lens, and to provide a unique 3Dcurved laser sheet upon exiting a race outlet. The output can also be2D. These 2D and 3D curved output scenarios can also be achieved with arace having a terminal end—or dead leg.

FIG. 29A depicts a toroidal full ring lens 2912 which has three portsconnecting to two races with hollow circular sections of radius a, wheretwo of the ports 2900, 2902 are inlets and the other an exit port 2904.The inlet ports 2900 and 2902 are configured to allow one or more laserbeams into the lens where firing time can be leveraged. FIG. 29Billustrates the internal race circuits of the lens from FIG. 29A, wherethe trace of the exterior lens boundary is shown as circular curvesalong with the bounding curves that formed the arbitrary ring section.Race 1 (2906) has the convergent inlet port 2900 that leads to thedivergent exit port 2904. Race 2 (2908) has the convergent inlet port2902 that leads to a divergent terminating end 2910 interior to thelens. The ports 2900, 2902 and 2904 and termination end have surface2610 that are generated from inclined lines that are revolved into conelike boundary surfaces. Moreover, cylindrical cuts are made nearest tothe lens boundary which preceded the cone shaped inlet ports 2900, 2902.These geometric features can control the electro-magnetic wave forms asthey traverse a particular lens. Race 1 (2906) traces a half circle,whereas Race 2 (2908) runs for a three-quarter turn while the circuitspirals gradually towards the lens center. Race 2 (2908) terminates atan end that had a wall perpendicular to the direction of wave travel,thus influencing an electromagnetic wave form traversing the race. Alsoon Race 2 (2908), by installing a widow in close proximity of the inletport so as to seal the hollow race to from a chamber, the balance of thehollow race volume could be filled with other materials which are fluid,gaseous, vapor, plasma and/or colloids. Chamber designs can be installedon Race 2 (2908) or on any hollow race. This leveraging of materials canbe applied to localized cavities that are embedded into a lens.

The surface on the terminating end of Race 2 (2908) is dielectric andoriented normal to the incident wave. It is known that for thisconfiguration a reflective wave will be generated due to the terminationsurface having a reflection coefficient, r, greater than zero and lessthan unity for the general case. Again, standing wave patterns can beexcited by combining the reflected wave with a part of the incident waveof equal amplitude. The balance of the incident wave gets transmittedinto the remainder of the lens medium while carrying electromagneticenergy with it. If the surface of the terminating end is layered with aperfectly conducting material, a standing wave pattern would also beproduced where the layer thickness is assumed sufficient to promote astanding wave. Under this scenario, a Poynting vector analysis showsthat the energy carried into the race by the incident wave is on averagecarried away by the reflected wave. Hence, no electromagnetic energywould be transmitted through the balance of the lens material if thesurface of the terminating end is layered sufficiently with a perfectconductor. In general, coatings and metallic material (layers) can beselectively deposited over lens surfaces, both internal and external toa lens.

FIG. 30 depicts a ring-like lens 3000 that has a dielectric material3004 on the outboard region and a metallic material 3020 on the inboardregion forming a composite lens. The inlet and exit ports 3006, 3008 areshown along with an incident ray 3010 and exit ray 3012. The section ofthis lens is chained together to form a D-shaped exterior boundary wherehalf of a circle is on the dielectric portion and including straightcurves with fillets on the metallic portion. A straight curve separatesthe two different materials at the interface.

Turning next to FIG. 31 , a trace of a lens section is included alongwith the internal features of the lens 3100. As can be seen, the hollowrace 3102 of section radius, a, is a circle relative to the lens 3100 inplan form, where both inlet and exit ports 3106, 3108 are convergent.Looking down the axis of revolution for any tangential location, boththe race 3102 and inlet port 3106 have two flat surfaces with twodiametrically opposed unit normal vectors (not shown) along the radialdirection mostly, while the exit port 3108 has one flat surface with aunit normal pointing along the inboard direction due to the exitdirection. The race 3102 has nine features in the form of half circulardisc like washers 3110 that are cut into the race on the inboard portionin a periodic fashion 20 degrees apart from one another. The dimensionsof the radial cut features 3110—relative to the metallic portion of thelens 3100—have lens material removed from the race surface at radius, a,to a depth into the lens material of radius R₁. These features 3110 havea relatively narrow gap thickness, g, cutout from the lens as well andthe gap is centered from the midpoint of the race by a distance, s. Theexpanded half circular disc feature 3110 of FIG. 31 shows star 3112representing the washer radial center.

For transverse magnetic waves (TM) traveling along the race, all of theelectromagnetic components considered are dependent on the global radialand tangential coordinates only. The electric field components ofsignificance act along the radial and tangential directions whereas themagnetic component of significance acts in the direction of therevolution axis. The differential form of Ampere's Law as it appeared inMaxwell's Equations indicated that a change in geometry of a conductingmedium which a magnetic field travels across can generate an electriccurrent. As a TM wave washed through the race from the inlet port itencountered a geometric expansion due to the gap cutout where themagnetic component excited a current as per Ampere's Law. This currentis shorted in the gap and the current also produced electric andmagnetic field components which are essentially constant over thegap—ignoring the higher order fringing fields at the gap edges.

In particular, the tangential electric field component over the gaplength, g, jumps from zero at the leading gap edge to a limiting valueE_(o) across the gap region and then back to zero as it reaches thetrailing gap edge where the race section became symmetric again,yielding a distribution that approximated one tooth of a periodic squarewave—neglecting edge effects. This pop in the electric field occurs atall of the gaps which are in periodic series with respect to themetallic race conductor thereby generating a periodic square wave. Bothforward traveling and backward traveling spatial harmonic waves areexcited, where the set of spatial harmonics are physically coupled toeach other by the periodic interaction between the boundary surfaces ofthe gap collection. The boundaries described by the race radius a, theperiodic spacing of the feature cutouts s, and the gap thickness g, andthe number of harmonics excited among other variables influence theelectric field components of the spatial harmonics, while the radial gapdepth R₁ among other variables influences the magnetic field componentof the spatial harmonics. The phase constant is inversely proportionalto the gap spacing s among other variables. This implies that for theset of backward and forward traveling spatial harmonic waves thecorresponding cutoff frequency, phase velocity, and group velocity canbe tuned to satisfy many service requirements. Likewise, theelectromagnetic field components can be tuned for a variety of servicerequirements. Other periodic features, similar to the half washerdisclosed herein, can excite and manage spatial harmonics ofelectromagnetic waves as well. The lens configuration of FIG. 31 depictshow a composite dielectric and metallic material combined with a raceand race features can be implemented to enhance the electromagnetic waveforms while traversing a lens.

As with all of the aforementioned race features, coatings can be appliedover all of the surfaces or portions judiciously selected thereof tomanage a laser light wave as it enters, traverses through, and exits alens. Similarly, localized coatings, films, and metallic coatings andmetallic layers can be deposited over regions of the lens surface.Stacked dielectric coating thickness dimensions can affect EM impedance.

Turning now to FIG. 32 , rings that are not full rings are referred toas ring segments or ring sectors. Ring 3200 illustrates a full ring,from FIG. 9 , swept from a general section along a tortuous path. Theradius, R, from the centroid 3202 of the ring 3200 to the centroid 3204of the ring section varies with path location. The ring centroid 3202 isidentified in ring 3200 and the section centroid 3204 is represented bythe dashed line passing through the centroid of the lens sections. Byreplacing the arbitrary lens section of ring 3200 with definitegeometry, practical lenses can be manufactured, where the dimensions aredependent on the service requirements. The angular tangential incrementΔΘ along the path swept out by the section represents a sector orsegment of the ring of arbitrary shape. Lens segments with segment angleΔΘ having practical sections that can be swept about any path where thesections are simply connected without cavities, simply connected withone cavity, and/or multiply connected. Composite lens segments 3300-3308are depicted in FIG. 33 , having metallic materials vii and dielectricmaterials viii. These lens segments 3300-3308 can be discrete lenssegments or can be representative chunks of lenses of full rings inother embodiments. Various of said segments 3300-3308 can be utilized toform a ring lens 3200 or similar ring lenses. Further these segments3300-3308 can have simply connected sections as in segments 3306 and3308, or multiply connected sections as in segments 3302, 3304.Additionally, the lens 3310 has a tapered wall thickness that forms aslot in the lens. In some embodiments, the metallic material may beinterchanged with a dielectric material different than that wrappedaround the original metallic. Similarly, other combinations of metallicand dielectric materials are envisioned within the scope of the presentdisclosure.

Turning last to FIGS. 34A and 34B, lens segments 3400 and 3402 are shownwith gratings 3404 or apertures 3406 therethrough to a central cavity.FIG. 34A illustrates a multiply connected lens segment with a coating orlayering, or by other means known to those with ordinary skill in theart selectively deposited or layered over the lens exterior forminggratings 3404. The gratings 3404 have a width, w, and are separated byspacing, s, as shown in the figure. The width and/or spacing may beuniform or non-uniform. Leaky laser light cannot pass through thegratings and can diffract into various diffraction orders. The gratings3404 shown in FIG. 34B can be thought of as providing an opposite effectas apertures. For example, the gratings 3404 can redirect leaky laserlight passing therethrough while the surrounding surface area (betweenthe grating) can emit redirected leaky laser light.

FIG. 34B illustrates an array of radiating apertures 3406, by coatingwith films or thin layering or by other means known to those withordinary skill in the art the exterior lens surface of lens segment3402. These apertures 3406 may be selectively placed onto the surface ofa lens. In the example of FIG. 34B, the apertures 3406 are on thesurface of a simply connected dielectric lens that has a crescentsection having one cavity which wrapped around a metallic material. Theapertures 3406 may be arranged uniformly or with non-uniform spacing,size, and/or shape. Leaky laser light can pass through the apertures3406 while the balance of the external surface does not facilitate thepassage of EM waves. As aperture size decreases, the radiation spreadingincreases. Thus, the geometry of these features can affect radiationoutput. For apertures, the radiation intensity emitted can be separatedinto two distinct zones. The near field zone is known as the Fresneldiffraction zone, while the far field is known as the Fraunhoferdiffraction zone.

In other embodiments, the features shown in FIG. 34B may be consideredgrids formed on the exterior lens surface. Such grids are akin to thegrating features. In other words, where apertures pass leaky laser light(and the remaining portion of the lens surface may block light passage),grids may take the form of apertures but block the passage of lighttherethrough (and thus the remaining portion of the lens may permit theleaky light passage).

While the features of FIG. 34A are shown with uniform spacing, size andshape, in other embodiments these features can be non-uniform in termsof spacing, size, and/or shape, and a mixture thereof. Similarly, whilethe features 3402 of FIG. 34B are shown with non-uniform spacing, sizeand shape, in other embodiments these features can be uniform in termsof spacing, size, and/or shape and a mixture thereof. These grating andaperture schemes, and grid schemes, can be applied to any lensconfiguration. Surface conditioning can be used to prepare the hostsurface for coating and layering of these features or surfaceconditioning can be used alone to produce and/or enhance similar outputeffects. Surface conditioning can be done by a plurality of etchingtechniques such as but not limited to microchemical etching, selectivelaser ablation, mechanical ablative/mechanical surface conditioningtechniques or other techniques known in the industry. The output effectsand/or enhancements include but are not limited to tailoring of theamplitude of the leaky light output and/or changing the apparent sheetthickness or width at selected positions across the span of the lasersheet.

The invention has been described with reference to the exampleembodiments described above. Modifications and alterations will occur toothers upon a reading and understanding of this specification. Exampleembodiments incorporating one or more aspects of the invention areintended to include all such modifications and alterations insofar asthey come within the scope of the appended claims and their equivalents.Finally, it should be understood that the figures herein are notnecessarily drawn to scale.

What is claimed is:
 1. A lens configured to transform an impinging laserbeam into a laser sheet, the lens comprising: an exterior surfacedefined by at least one curve rotated along a path about a centroid ofthe lens; and at least one section having an outer perimeter defined bythe at least one curve, and a cavity defined by an inner surface of thelens, wherein: the lens is configured to transform the laser beam into a2D laser sheet or a 3D laser sheet based on a relative incident angle ofthe laser beam on the lens, the lens is a prism, ring, disc, plate,shell, or solid volumes, and the laser sheet has a fan angle up to andincluding 360 degrees based on an incident location of the laser beam onthe lens, a distance between a source of the laser beam and the centroidof the lens, and a distance between the centroid of the lens aprojection surface of the laser sheet.
 2. The lens according to claim 1,wherein the plurality of curves are chained together to define theexterior surface of the lens.
 3. The lens according to claim 1, whereinthe lens has a Dupin cycloid shape.
 4. The lens according to claim 1,wherein at least a portion of the inner surface or the exterior surfaceof the lens is coated, has a deposition thereon, or is of a materialdifferent from a base material of the lens.
 5. The lens according toclaim 1, wherein at least a portion of the inner surface or an inboardregion of the exterior surface comprises a reflective coating.
 6. Thelens according to claim 1, wherein the at least one section is at leastpartially metallic, and wherein a base material of the lens is adielectric.
 7. The lens according to claim 1, wherein at least a portionof the exterior surface of the lens comprises a coating selectivelydeposited thereon so as to form gratings, grids, or apertures on theexterior surface.
 8. The lens according to claim 7, wherein theplurality of gratings or apertures are spaced apart and/or sizednon-uniformly.
 9. The lens according to claim 1, further comprising arace.
 10. The lens according to claim 1, further comprising a solid raceextending about the exterior surface of the lens and projecting from theexterior surface of the lens.
 11. The lens according to claim 1, whereinthe at least one section is simply connected.
 12. The lens according toclaim 1, wherein the at least one section is multiply connected.
 13. Alens configured to transform an impinging laser beam into a laser sheet,the lens comprising: an exterior surface defined by at least one curverotated along a path about a centroid of the lens; and a race extendingthrough the lens between an inlet port at the exterior surface and anoutlet port at the exterior surface or a terminating end within thelens, the race being configured to allow a portion of the laser beam topass therethrough from the inlet port to the outlet port, wherein: thelens is configured to transform the laser beam into a 2D laser sheet ora 3D laser sheet based on a relative incident angle of the laser beam onthe lens, and the laser sheet has a fan angle up to and including 360degrees.
 14. The lens according to claim 13, wherein the lens is aprism, ring, disc, plate, shell, or solid volume.
 15. The lens accordingto claim 14, wherein the fan angle is based on an incident location ofthe laser beam on the lens, a distance between a source of the laserbeam and the centroid of the lens, and a distance between the centroidof the lens and a projection surface of the laser sheet.
 16. The lensaccording to claim 13, further comprising a solid race extending aboutthe exterior surface of the lens and projecting from the exteriorsurface of the lens.
 17. The lens according to claim 13, wherein therace comprises a plurality of sections having different shapes.
 18. Thelens according to claim 13, wherein the race is at least partiallyfilled with a solid, fluid, gas, colloid, plasma, and/or vapor.
 19. Thelens according to claim 13, comprising a plurality of races extendingthrough the lens between a plurality of respective inlet and outletports and/or inlet ports and terminating ends.
 20. The lens according toclaim 13, wherein the race splits into a plurality of paths at a firstjunction, and reforms into a single path at a second junction.